sciencemag.org SCIENCEIMAGE: SIYU WU AND YUGANG SUNBy Siyu Wu and Yugang SunT
essellating tiles and bricks of well-
defined geometric shapes can be traced
back to the ancient age (~4000 BCE),
when Sumerians used clay tiles to dec-
orate building walls. Since 1619 when
Johannes Kepler first wrote about tes-
sellation, it has become a topic of interest in
disciplines including pure mathematics, ma-
terials science, chemistry, art, architecture,
and industrial design. Packing convex po-
lygonal shapes into three-dimensional (3D)
Euclidean space was conjectured by Ulam to
form a more dense tessellation than spheres,
but that conjecture was theoretically verified
only recently for packing tetrahedrons in a
quasicrystalline (QC) arrangement through
shape-induced entropic interactions ( 1 ).
Forming a QC superlattice of tetrahedrons
with rotational symmetricity yet no transi-
tional periodicity ( 2 ) has represented a grand
experimental challenge. On page 1396 of
this issue, Nagaoka et al. ( 3 ) report a “flex-ible polygon tiling rule” that guided the self-
assembly of nanometer-sized tetrahedrons of
cadmium chalcogenide (CdSe-CdS core-shell
nanoparticles) to form a 10-fold QC superlat-
tice at a liquid-air interface (see the figure).
The authors synthesized tetrahedral
semiconductor nanocrystals (NCs) with two
different types of surfaces, including three
equivalent {10 ̄11} facets (A type) and one
{0002} facet (B type), which were covered
with oleic acid and octadecylphosphonic
acid, respectively. Such differentiation of
surface chemistry promoted the selective
facet-to-facet alignment and attachment
(i.e., A-to-A and B-to-B) of neighboring tet-
rahedrons as they moved close together in
the course of self-assembly. The preferred
alignment of tetrahedral NCs led to the for-
mation of decagon units with 10-fold sym-
metry, each of which contained a tetrahelix
ring ( 4 ) surrounding a vertically stacked
pair of tetrahedrons in the center.
Unlike clay bricks bonded with mortar,
the tetrahelix rings of the decagon units
were flexible, which allowed the entire sys-
tem to gain additional stability through lo-
cal configuration changes at the interfacesof packed decagon units ( 5 ). The flexible
polygon tiling rule facilitated the tessella-
tion of the decagon units into 10-fold QC
superlattices of the nanosized tetrahe-
drons. This rule can be generalized to pack
any polygon units with even numbers of
edges n—for example, octagons, decagons,
dodecagons, and tetradecagons—forming
3D QC tessellations. During packing, two
neighboring polygons may overlap with
only one or two edges. The two polygons
need to remain intact when there is no edge
or only one edge overlapping; when two
edges overlap, the overlapped edges need
to become flexible and transform into one
straight edge (the flexible edge). Each poly-
gon must have at least one flexible edge and
at most n/2 flexible edges, and the gener-
ated tessellation pattern must have a QC
order with an n-fold rotational symmetry.
When packing tetrahedral solids, exter-
nal compressive force determines the pack-
ing order and density ( 6 ). Applying stronger
compression forces to the tetrahedral NCs
in the course of self-assembly prevented
the formation of the tetrahelix polygon
rings and QC tessellation at the liquid-airQUASICRYSTALSTessellating tiny tetrahedrons
A tiling rule guides the formation of quasicrystalline superlattices of nanocrystals
Department of Chemistry, Temple University, Philadelphia, PA
19122, USA. Email: [email protected]INSIGHTS
PERSPECTIVES
1354 21 DECEMBER 2018 • VOL 362 ISSUE 6421
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